which also arise in applications; see e.g. the Hénon map in [1]} or the detailed analysis of invariant manifolds and corresponding foliations in 2-dimensional maps (REF ) in [2]}. Generically, the maps (REF ) are regularly (as opposed to singularly) perturbed in the sense of Definition REF , since the critical set \(\lbrace (O_k^{\textnormal {T}},y) : f(O_k^{\textnormal {T}},y,0) = O_{n-k} \rbrace \)  is generically empty or comprised entirely of isolated points \((O_k^{\textnormal {T}},y_\ast )\)  for which \(y_\ast \)  satisfies \(\tilde{f}(O_k^{\textnormal {T}},y_\ast ,0) = O_{n-k}\) . It follows that the theory developed herein does not in general apply for maps in the form (REF ), since they may not satisfy Assumption REF .

